What is the difference between the methods for calculating volume in calculus like disk,washer,cross section, shell, etc?
In: Mathematics
All volume formulas are doing the same thing…slice the volume up into very tiny pieces that have a “simple” formula for their volume, then add them all up.
The difference between the different methods is what small shape you’re using. The choice is arbitrary to the end result but how easy it is to do the math depends on matching the small volumes to the big one. Hole in the middle? Washers. Cylindrical? Disks. Spherically symmetric? Shells. Arbitrary? Cross sections (hopefully sliced in a direction that gives you nice cross sections). They’re all about choosing the setup to make the intermediate steps simpler so the path to the end is easier.
Edit: it’s entirely possible to derive the correct formula for the volume of a sphere using small cubes…it just sucks. The geometry is horrible. If you use tiny shells it’s really easy.
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